Gerd Faltings Proves Mordell's Conjecture (1983)
Gerd Faltings Proves Mordell's Conjecture (1983)
Overview
German mathematician Gerd Faltings (1954 ) proved Mordell's conjecture in 1983, an accomplishment that earned him the prestigious Fields Medal, mathematics' highest honor. His method of altering a familiar geometric theorem into algebraic terms led him to solve the complex geometric theorem proposed by Louis Mordell in 1922. His success in proving this conjecture has contributed to the advancement of the studies of algebra and geometry.
Background
Faltings has been awarded many honors in his lifetime. Most notable is his receipt of the distinguished Fields Medal, which he received in 1986. Faltings earned this honor because he proved Mordell's conjecture using algebraic geometry. The conjecture that Louis Mordell (18881972) initiated in 1922 stated that a given set of algebraic equations with rational coefficients defining an algebraic curve of n greater than or equal to 2 must have only a finite number of rational solutions. To prove the conjecture, Faltings used a method initiated out of developing the Arakelov theorem in order to produce an arithmetic version of yet another theorem—the commonly used, geometrically based RiemannRoch theorem.
Mordell in his era proved the finite generation in mathematics, and mathematicians have since built upon his findings and developed the crossover use of algebra in solving geometric problems utilizing the methods of one to enable proof of the other. André Weil (19061998), for example, extended Mordell's result of finiteness using algebraic geometry to prove finiteness in Abelian varieties over number fields using multiple geometric strategies. Geometrically, Mordell's conjecture translated well into algebraic terms, providing scholars with the opportunity to use functional algebraic methods to solve geometric problems. Mordell's conjecture, when translated algebraically, revealed that a group of curves based on his conjecture would result in only a finite number of sections if the group were not constant. Thus, Faltings was able to prove the accuracy of the conjecture.
Impact
Faltings's proof is important for several reasons, which are listed below and then discussed in greater detail in the paragraphs following:
 It shows how the work of one mathematician can lead others to advance the knowledge of a given field.
 It has led others to find new ways to solve ageold problems related to number theory.
 It was instrumental in helping Andrew Wiles finally prove what was probably mathematics' greatest unsolved problem: Fermat's last theorem.
The historical context of Faltings's achievement is important because it sheds light on how the work of one mathematician can lead other scholars to advance knowledge in a given area. A mathematician named GilletSoule, for instance, expanded on the technique of utilizing many arithmetic surfaces to translate another wellknown theorem, the ReimannRoch theorem. GilletSoule also added varieties pertaining to arbitrary dimensions, complex differential geometry as it relates to the components at infinity, and real partial differential equations. He utilized this expansion of the ReimannRoch theorem by using both the HirzebruchGrothendieck theorem and the original RiemannRoch theorem. Another example of how scholars have built upon the accomplishments of others outside of their specialty in mathematics can be seen in Bismut's work. Bismut took GilletSoule's developments, which sprung from Faltings's work as it related specifically to real partial differential equations, and expanded them to function with the analogues of Green functions, particularly in higher dimensional cases. Following Faltings's proof of Mordell's conjecture, another mathematician—a man named P. Vojta—was later able to find another way to prove Mordell's conjecture. To accomplish this, he utilized the fundamental information that proved the ReimannRoch theorem, and then he expounded upon particular aspects of it to create yet another way to prove Mordell's conjecture.
All of these various advancements from great scholars such as Faltings encompass the expansion of several areas of mathematics that have found their residence in the RiemannRoch theorem. Throughout time, the RiemannRoch theorem has been transformed into something that encompasses other arithmetic areas of study as a result of dedicated scholars who have contributed to the discovery of new elements in algebra and geometry, adding new dimensions to original theorems. Faltings has written a book (Lectures on the Arithmetic RiemannRoch Theorem) discussing the RiemannRoch theorem in algebraic terms.
The widespread use of contemporary methods to solve mathematical problems posed ages ago relative to number theory, as is the case with Faltings's proof of Mordell's conjecture, has led many scholars to discover new ways to incorporate several areas of mathematics to solve old problems. In his case, Faltings was able to utilize the general concepts of algebraic geometry along with complete objects, including the components of Arakelov's infinity. In using the knowledge of Arakelov's infinity theorem, Faltings illustrated the importance of building upon new discoveries. The Arakelov theory prepared the means for other mathematicians to create unification of intersection theory in algebraic fields as well as unification in classical intersection theory. Arakelov defined the infinity of intersection numbers as the values specifically of Green's functions.
Proving Mordell's conjecture led Faltings to further the study of Fermat's last theorem, long considered the greatest unsolved problem in mathematics until it was finally proven by Andrew Wiles (1953 ) in 1994. Faltings illustrated that Fermat's last theorem, x^{n} + y^{n} = z^{n}, is only capable of a finite number of rational solutions in integers for which n is greater than 2 and, therefore, has no solutions. Faltings also used ideas of Vojta's to prove a conjecture created by Serge Lang that dealt with higher dimensional diophantine analogues for use in basic subvarieties of Abelian varieties. Diophantine approximations are the formal relationships existing in the functions of a theory and the counting processes used to determine the number of functions in classical as well as specific asymptotic estimates.
Faltings's methods of algebraic geometry have also been used to examine the particular finiteness conjecture in Galois representations. The practicality of utilizing various areas in mathematics has given rise to many developments in the various fields. Interest in mathematics has grown tremendously with the advancements accomplished in number theory throughout time.
BROOKE COATES
Further Reading
Faltings, Gerd. Lectures on the Arithmetic RiemannRochTheorem (Anal of Mathematics Studies, 127). Princeton, NJ: Princeton University Press, 1992.
Faltings, Gerd. Rational Points. Friedrich Vieweg & Sohn, 1992.
Soule, C., D. Abramovich, J. K. Kramer, and J. F. Burnol. Lectures on Arakelov Geometry (Cambridge Studies in Advanced Mathematics, 33). New York: Cambridge University Press, 1995.
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Gerd Faltings Proves Mordell's Conjecture (1983)." Science and Its Times: Understanding the Social Significance of Scientific Discovery. . Encyclopedia.com. 20 Sep. 2018 <http://www.encyclopedia.com>.
"Gerd Faltings Proves Mordell's Conjecture (1983)." Science and Its Times: Understanding the Social Significance of Scientific Discovery. . Encyclopedia.com. (September 20, 2018). http://www.encyclopedia.com/science/encyclopediasalmanacstranscriptsandmaps/gerdfaltingsprovesmordellsconjecture1983
"Gerd Faltings Proves Mordell's Conjecture (1983)." Science and Its Times: Understanding the Social Significance of Scientific Discovery. . Retrieved September 20, 2018 from Encyclopedia.com: http://www.encyclopedia.com/science/encyclopediasalmanacstranscriptsandmaps/gerdfaltingsprovesmordellsconjecture1983
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.